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# Numerical Study of Temperature-Dependent Viscosity and Thermal Conductivity of Micropolar Ag–MgO Hybrid Nanofluid over a Rotating Vertical Cone

1 Department of Mathematics, University of Eswatini, Kwaluseni, M201, Eswatini

2 Department of Mathematical Sciences, Sol Plaatje University, Kimberley, 8301, South Africa

3 Department of Mathematical and Computational Sciences, University of Venda, Thohoyandou, 0950, South Africa

* Corresponding Author: Mekonnen S. Ayano. Email:

(This article belongs to the Special Issue: Computational and Numerical Advances in Heat Transfer: Models and Methods I)

*Frontiers in Heat and Mass Transfer* **2024**, *22*(4), 1153-1169. https://doi.org/10.32604/fhmt.2024.048474

**Received** 08 December 2023; **Accepted** 29 January 2024; **Issue published** 30 August 2024

## Abstract

The present paper examines the temperature-dependent viscosity and thermal conductivity of a micropolar silver ()−Magnesium oxide () hybrid nanofluid made of silver and magnesium oxide over a rotating vertical cone, with the influence of transverse magnetic field and thermal radiation. The physical flow problem has been modeled with coupled partial differential equations. We apply similarity transformations to the non-dimensionalized equations, and the resulting nonlinear differential equations are solved using overlapping grid multidomain spectral quasilinearization method. The flow behavior for the fluid is scrutinized under the impact of diverse physical constraints, which are illustrated graphically. The results of the skin friction coefficient and Nusselt number varying different flow parameters are presented in the form of a table. It is observed that the main flow of the hybrid nanofluid, nano particle fraction of silver and Magnesium/water, enhances compared to the mono-nano fluid as the coupling number increases. The application of studies like this can be found in the atomization process of liquids such as centrifugal pumps, viscometers, rotors, fans.## Keywords

Thermal properties of most base fluids used in industry are low and do not address the needs of recent technology. Nowadays, effective active heat transfer methods like ultrasonic vibration have been explored by Delouei et al. [1]. Another mechanism involves adding nanoparticles to the base fluid to improve heat transfer; this approach, discussed by Siavashi et al. [2], utilizes nanoparticles, known as nanofluids, to overcome this deficiency. At present, a widely explored research area for enhancing the thermal conductivity of base fluids is nanotechnology, as investigated by Izadi et al. [3]. The nanofluids have widespread applications in various engineering and industrial processes. Applications of this nature encompass cooling systems, heat exchangers, geothermal and solar energy systems, and biomedicine, particularly in the realm of targeted drug delivery, as highlighted by Siavashi et al. [2]. A considerable number of researchers have discussed the flows of nanofluid and its applications [2–7]. Akmal et al. [8] observed that the heat transfer rate increases as the values of the thermophoresis parameter increase. Zeeshan et al. [9] analyzed a convective MHD nanofluid water/ethylene glycol based flow towards a vertical cone. Hady et al. [10] reported a rapid reduction in heat transfer rates due to an increase in the nanofluid’s temperature and the solid volume fraction. Liu et al. [11] proposed a

Nanoparticles such as

The behaviour of a hybrid nanofluid composed of

It is known that many of the industrially and technologically important fluids are non-Newtonian fluids. In this article, a non-Newtonian micropolar fluid is considered. The most common type of non-Newtonian micropolar fluid is a fluid with a micro-structure that may consist of rigid, randomly oriented particles suspended in a viscous medium. MHD flow and heat transfer of carbon nanotube (

Motivated by the aforementioned works, the present study extends the work of Malik et al. [29]. This extension is not a simple one as the authors aim to address a hybrid nanofluid with spin gradient viscosity for a non-Newtonian micropolar fluid, an aspect that was not covered in their study. Furthermore, this extension addresses a gap not explored in classical fluid dynamics, as far as our knowledge extends. We examine the effects of volumetric nanofractions, micropolar, and viscous fluids and compare their impacts.

2 Mathematical Formulation of the Problem

Consider an inverted isothermal rotating cone with a semi-vertical angle of

where

where

According to Khan et al. [33], volume fraction, thermal conductivity, density, heat capacitance, thermal expansion coefficient, thermal conductivity and viscosity of the hybrid nanofluid, are defined by Eqs. (8)–(14), respectively:

Using Eqs. (6), (8)–(14) and the transformations specified by [29]:

the system of partial differential Eqs. (1)–(5) is transformed into ordinary differential Eqs. (15)–(18):

where

The nondimensional boundary conditions take the form:

The skin friction coefficient and local Nusselt number are defined as:

where

The corresponding non-dimensional form for Eqs. (20) and (21) are as follows:

where

Numerical solutions to the non-dimensionalized system of Eqs.(15)–(18) are obtained using the OGMDSQLM, proposed by Mkhatshwa [34]. The numerical solutions were generated through codes developed in MATLAB. The solution algorithm of this numerical method is based on numerical techniques namely QLM, spectral collocation, multidomain overlapping grid and Lagrange interpolation polynomials with Gauss-Lobatto-Chebyshev grid points. To implement the method, the integration domain,

Using the SQLM method proposed by Motsa et al. [35], the coefficients of the unknowns are evaluated at the previous iteration level

As mentioned earlier, the essence of the overlapping grid multidomain spectral quasilinearization is the slicing of the physical domain into a finite number of overlapping subintervals, the non-dimensional boundary conditions then become as written in Eq. (30).

The linearized system of Eqs.(24)–(27) can be written compactly as shown in system (31), where

The system of Eq. (31) can be represented in matrix from as system (32). The column vectors of unknowns at a subinterval and current iteration level are:

where

and

Applying the pseudospectral method at the boundaries gives:

The OGMDSQLM leads to a sparse form of the coefficient matrix in system (32), which is expressed in its expanded form below, with the boundary conditions incorporated accordingly on the diagonal submatrices.

To begin the iterative search for approximate solutions, the following functions:

that satisfy the boundary conditions are taken as approximate solutions.

To check the validity of the current numerical scheme, a comparison of the present values of skin friction and Nusselt number with those published in the literature by Malik et al. [29], Chamkha et al. [36] is presented in Table 2. Using the nanoparticle volume fractions of hybrid nanofluid consisting of portions of nano particles

Fig. 4a shows that the smaller the variable viscosity parameter, the more the primary velocity diminishes. Physically, this implies that in the higher viscous region (near the boundary), the dominant force is buoyancy, and an increase in nanofluid concentration leads to an increase in velocity. The more hybrid nanofluid, the higher the thermal conductivity, which allows an increase in Brownian motion and, consequently, particle energy transfer. It is also interesting to note that away from the vicinity of the boundary, in the free stream region, the reverse effect is observed. However, it is evident from Fig. 4b that an increase in the variable viscosity parameter results in a decreasing effect on the secondary flow. This is because as the coupling number K increases, the fluid becomes more micropolar, and consequently, the particle sniping effect significantly retards the secondary flow. For the fractional volume of the hybrid nanofluid,

Fig. 5a,b shows the impact of porosity parameter on both primary and secondary flows. It is clear from the figures that primary velocity decreases with an increase in the porosity parameter. The velocities of the hybrid nanofluid,

In Tables 3 and 4, the flow behaviour for different values of radiation and magnetic parameter on local sikin friction effect and Nusselt number are presented. The cases for mono-nanofluids

A numerical investigation for unsteady magneto micropolar

• Micropolar fluid assists the secondary flow and diminishes the main flow. The water hybrid nanofluid,

• The rotational effect for the hybrid nanofluid near the boundary is greater than that of the mono-nanofluid.

• For the water hybrid nanofluid,

• The variable viscosity parameter reduces the temperature for the hybrid nanofluid,

• The porosity parameter decreases the main flow; however, the inclusion of the hybrid nanofluid

Acknowledgement: The authors would like to thank the anonymous reviewers for their valuable comments.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: The authors confirm contribution to the paper as follows: conceptualisation: Mekonnen S. Ayano; writing original draft: Mekonnen S. Ayano; methodology, code writing: Thokozani N. Khumalo, Stephen T. Sikwila; Matlab code creation and validation: Mekonnen S. Ayano, Thokozani N. Khumalo, Stephen T. Sikwila; analysis and interpretation of results: Mekonnen S. Ayano, Stanford Shateyi; final manuscript preparation: Mekonnen S. Ayano, Stephen T. Sikwila; validation of results: Mekonnen S. Ayano, Stanford Shateyi. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The readers can contact the corresponding author and request the Matlab codes used to generate the data.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

**APA Style**

*Frontiers in Heat and Mass Transfer*,

*22*

*(4)*, 1153-1169. https://doi.org/10.32604/fhmt.2024.048474

**Vancouver Style**

**IEEE Style**

*Front. Heat Mass Transf.*, vol. 22, no. 4, pp. 1153-1169, 2024. https://doi.org/10.32604/fhmt.2024.048474

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